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Theorem ablgrp 13046
Description: An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
Assertion
Ref Expression
ablgrp  |-  ( G  e.  Abel  ->  G  e. 
Grp )

Proof of Theorem ablgrp
StepHypRef Expression
1 isabl 13045 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
21simplbi 274 1  |-  ( G  e.  Abel  ->  G  e. 
Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148   Grpcgrp 12831  CMndccmn 13041   Abelcabl 13042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-abl 13044
This theorem is referenced by:  ablgrpd  13047  ablinvadd  13066  ablsub2inv  13067  ablsubadd  13068  ablsub4  13069  abladdsub4  13070  abladdsub  13071  ablpncan2  13072  ablpncan3  13073  ablsubsub  13074  ablsubsub4  13075  ablpnpcan  13076  ablnncan  13077  ablnnncan  13079  ablnnncan1  13080  ablsubsub23  13081
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